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%% groceryListOfCats.tex
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%% Made by Alex Nelson
%% Login   <alex@tomato>
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%% Started on  Fri Jun 19 12:24:01 2009 Alex Nelson
%% Last update Fri Jun 19 12:24:01 2009 Alex Nelson
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We have categorical description of the various different
mathematical objects, here's some selected examples:
\begin{enumerate}
\item[(Concrete)] If the category's objects are ``sets with
  structure'' (e.g. $\ms{Mon}$ has monoids, $\ms{Top}$ has
  topological spaces, etc.), then the category is ``concrete''. 
\item[(Discrete Categories)] The only morphisms are identity
  morphisms.
\item[(Groups)] If the category is a monoid with the extra
  condition the morphisms are invertible.
\item[(Monoids)] The category has just one object, so it's completely
  determined by its morphisms.
\item[(Thin)] If there is at most one morphism from an object to another.
\end{enumerate}

Here we simply give a grocery list of categories that common come
up. The following are commonly used categories, given by their
shorthand notation commonly used in the literature.
\begin{description}
\item[0] is the empty category (no objects, no arrows).
\item[1] is the singleton category, with one object and only one
  arrow -- the identity morphism.
\item[2] is the category with two objects $a,b$ and exactly one
  morphism $a\to b$ which is not the identity.
\item[3] is the category with three objects $a,b,c$ and exactly
  three morphisms $a\to{}b$, $b\to{}c$, $c\to{}a$, none identities.
\item[$\downdownarrows$] is the category with two objects $a,b$ and
  two morphisms $a\rightrightarrows{}b$ none identities.
\item[Mat$_\text{K}$] Matrices over a commutative ring $K$, we
  have objects be positive integers $m,n,\ldots$ and each
  $m\times n$ matrix $A$ eats in a vector in $K^n$ and spits out
  a vector in $K^m$, so it's a morphism $A:n\to m$.
\item[Set] Objects are all small sets, and morphisms are
  functions between them.
\item[Cat] Objects are all small categories, and morphisms are
  functors.
\item[Mon] Objects are small monoids, morphisms are morphisms of monoids.
\item[Grp] Objects are small groups, morphisms are morphisms of groups.
\item[Ab] Objects are small Abelian groups, morphisms are
  morphisms between them.
\item[Rng] Objects are small rings, morphisms are morphisms of rings.
\item[CRng] Objects are small commutative rigns, morphisms are
  morphisms between them.
\item[$R$-Mod] All small left modules over the ring $R$, with
  linear maps.
\item[Mod-$R$] Small right $R$-modules.
\item[Top] Small topological spaces and continuous maps.
\end{description}
